# Time adaptive Zassenhaus splittings for the Schr\"odinger equation in the semiclassical regime

## TL;DR

This paper introduces a novel time-adaptive Zassenhaus splitting method for solving the Schrödinger equation in the semiclassical regime, effectively handling high oscillations and improving computational efficiency.

## Contribution

It combines asymptotic Zassenhaus splitting with time adaptivity, enhancing efficiency and accuracy in numerical solutions of semiclassical Schrödinger equations.

## Key findings

- Method reduces computational cost in high oscillation regimes
- Time adaptivity optimizes step size during simulations
- Numerical examples demonstrate improved accuracy and efficiency

## Abstract

Time dependent Schr\"odinger equations with conservative force field U commonly constitute a major challenge in the numerical approximation, especially when they are analysed in the semiclassical regime. Extremely high oscillations originate from the semiclassical parameter, and call for appropriate methods. We propose to employ a combination of asymptotic Zassenhaus splitting with time adaptivity. While the former turns the disadvantage of the semiclassical parameter into an advantage, leading to highly efficient methods with low error constants, the latter enables to choose an optimal time step and to speed up the calculations when the oscillations subside. We support the results with numerical examples.